The transition from conservative to dissipative flows in class-B laser model with fold-Hopf bifurcation and coexisting attractors

2021 ◽  
Author(s):  
Yue Li ◽  
Zengqiang Chen ◽  
Mingfeng Yuan ◽  
Shijian Cang
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Blow Up ◽  
Laser System ◽  
Stable State ◽  
Phase Volume ◽  
External Interference ◽  
Dynamical Behaviors ◽  
Special Cases ◽  
Class B

Abstract Dynamical behaviors of a class-B laser system with dissipative strength are analyzed for a model in which the polarization is adiabatically eliminated. The results show that the injected signal has an important effect on the dynamical behaviors of the system. When the injected signal is zero, the dissipative term of the class-B laser system is balanced with external interference, and the quasi-periodic flows with conservative phase volume appear. And when the injected signal is not zero, the stable state in the system is broken, and the attractors (period, quasi-period, and chaos) with contractive phase volume are generated. The numerical simulation finds that the system has not only one attractor, but also coexisting phenomena (period and period, period and quasi-period) in special cases. When the injected signal passes the critical value, the class-B laser system has a fold-Hopf bifurcation and exists torus ”blow-up” phenomenon, which will be proved by theoretical analysis and numerical simulation.

scholarly journals Stability and bifurcation analysis of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response

2021 ◽  
Vol 18 (6) ◽  
pp. 7877-7918
Author(s):  
Shuangte Wang ◽  
◽  
Hengguo Yu ◽  
◽  
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Functional Response ◽  
Equilibrium Points ◽  
Critical Threshold ◽  
Predator Prey ◽  
Dynamical Behaviors ◽  
Stability And Bifurcation ◽  
Threshold Conditions ◽  
Existence And Stability

<abstract><p>In the paper, stability and bifurcation behaviors of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response are studied theoretically and numerically. Mathematical theory works mainly give some critical threshold conditions to guarantee the existence and stability of all possible equilibrium points, and the occurrence of Hopf bifurcation and Bogdanov-Takens bifurcation. Numerical simulation works mainly display that the Bazykin's predator-prey ecosystem has complex dynamic behaviors, which also directly proves that the theoretical results are effective and feasible. Furthermore, it is easy to see from numerical simulation results that some key parameters can seriously affect the dynamic behavior evolution process of the Bazykin's predator-prey ecosystem. Moreover, limit cycle is proposed in view of the supercritical Hopf bifurcation. Finally, it is expected that these results will contribute to the dynamical behaviors of predator-prey ecosystem.</p></abstract>

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

Turing Instability and Hopf Bifurcation in Cellular Neural Networks

2021 ◽  
Vol 31 (08) ◽  
pp. 2150143
Author(s):  
Zunxian Li ◽  
Chengyi Xia
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Turing Instability ◽  
Cellular Neural Networks ◽  
Bifurcation Theorem ◽  
Decoupling Method ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Approximate Expressions ◽  
Zero Equilibrium

In this paper, we explore the dynamical behaviors of the 1D two-grid coupled cellular neural networks. Assuming the boundary conditions of zero-flux type, the stability of the zero equilibrium is discussed by analyzing the relevant eigenvalue problem with the aid of the decoupling method, and the conditions for the occurrence of Turing instability and Hopf bifurcation at the zero equilibrium are derived. Furthermore, the approximate expressions of the bifurcating periodic solutions are also obtained by using the Hopf bifurcation theorem. Finally, numerical simulations are provided to demonstrate the theoretical results.

scholarly journals From chaos to stability in soliton mode-locked fibre laser system

2021 ◽  
Vol 67 (6 Nov-Dec) ◽  
Author(s):  
Morteza A. Sharif ◽  
K. Ashabi
Keyword(s):  
Laser System ◽  
Control Process ◽  
Stable State ◽  
Chaotic Regime ◽  
Fibre Laser ◽  
Control Parameters ◽  
Single Side ◽  
Encryption And Decryption ◽  
Encryption Decryption ◽  
Unstable States

Intracavity energy rate in a soliton mode-locked fibre laser is derived by solving the Haus master equation. The influence of net gain, absorber response, saturation energy, nonlinearity and absorption are investigated on stable/unstable states. Intracavity modes include the zeroth, first and higher order solitons. Accordingly, chaotic regime as well as breather modes is recognized as a conventional intracavity state. However, tuning the control parameters also results in a reverse bifurcation and thus returning to a stable state. Accordingly, a chaos-based encryption/decryption system is proposed taking the advantage of using a single-side control process; both the encryption and decryption procedures can be achieved by one of the actions of increasing/decreasing the control parameters.

scholarly journals Hopf Bifurcation and Chaos of a Delayed Finance System

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Xuebing Zhang ◽  
Honglan Zhu
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Center Manifold ◽  
Chaotic State ◽  
Bifurcation And Chaos ◽  
Finance System ◽  
Center Manifold Theorem ◽  
Normal Form Method ◽  
Simulation Results ◽  
The Stability

In this paper, a finance system with delay is considered. By analyzing the corresponding characteristic equations, the local stability of equilibrium is established. The existence of Hopf bifurcations at the equilibrium is also discussed. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical simulation results show that delay can lead a stable system into a chaotic state.

scholarly journals Rich Dynamics of a Brucellosis Model with Transport

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Juan Liang ◽  
Zhirong Zhao ◽  
Can Li
Keyword(s):  
Numerical Simulation ◽  
Sensitivity Analysis ◽  
Infectious Diseases ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Great Influence ◽  
Stable State ◽  
Initial Values ◽  
Si Model ◽  
The Basic Reproduction Number

Brucellosis is one of the major infectious diseases in China. In this study, we consider an SI model of animal brucellosis with transport. The basic reproduction number ℛ0 is obtained, and the stable state of the equilibria is analyzed. Numerical simulation shows that different initial values have a great influence on results of the model. In addition, the sensitivity analysis of ℛ0 with respect to different parameters is analyzed. The results reveal that the transport has dual effects. Specifically, transport can lead to increase in the number of infected animals; besides, transport can also reduce the number of infected animals in a certain range. The analysis shows that the number of infected animals can be controlled if animals are transported reasonably.

Analysis on Hopf Bifurcation and Complexity of one Kind Financial System

2011 ◽  
Vol 48-49 ◽  
pp. 813-816 ◽  
Author(s):  
Qi Zhang ◽  
Jun Hai Ma
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Dynamic Analysis ◽  
Dynamic Systems ◽  
Financial System ◽  
The Relationship ◽  
At Home

From a mathematical model of one kind complicated financial system, we make a dynamic analysis on this kind of system on the basis of studies of scholars both at home and abroad. We find characteristics of various dynamic systems driven by different parameters, and study possible Hopf bifurcation as well as the relationship between Hopf bifurcation and the values of parameters. Besides, we make use of algorithm to analyze complexity of the system. The results of numerical simulation prove that the theory used in the thesis is correct. This study is regarded with good theoretical and practical value.

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